Now [1]} implies that \(\nabla _p Q(p, d) = -\hat{\lambda }(p, d)\) , which is continuously differentiable from Lemma REF . Therefore, \(\nabla ^2_{pp} Q\)  is continuous and [2]} ensures the existence of \(\rho _2 > 0\)  such that \(\nabla ^2_{pp} Q(0, 0) + \frac{\rho }{2} \mathrm {I}\)  is positive definite and that its minimum eigenvalue is bounded strictly away from 0 for all \(\rho > \rho _2\) .
By a similar argument as in the proof of Lemma REF ,
continuity of \(\nabla ^2_{pp} Q\)  implies that there exists \(\epsilon _4^{\prime } > 0\)  (independent of \(\rho \) ) such that \(\nabla ^2_{pp} Q(p, d) + \frac{\rho }{2} \mathrm {I}\)  is positive definite, and hence the function \(F_d: p \mapsto Q(p, d) + \frac{\rho }{4} \left\Vert p \right\Vert ^2\)  is convex, whenever \((p, d) \in B_{\epsilon _4^{\prime }}(p, d)\) . Now fix \(\epsilon _4 = \min \lbrace \epsilon _4^{\prime }, \epsilon _1\rbrace  > 0\)  and \(d = d^k\) .
Convexity of \(F_{d_k}\)  implies:
\(F_{d_k}(p^k) + (0 - p^k)^\top \nabla F_{d_k}(p^k) \le F_{d_k}(0).\) 
